1. family and parenting
2. children

Demographic competition and inequality
aversion
Charles Roddie, Cambridge University∗
March 19, 2015
Abstract
If individuals care the number of their descendents, they will display relative preferences, and aversion to inequality: if fertility of others is reduced, or
distributed more equally, an individual’s descendents will face less competition.
Lower social mobility leads to greater aversion to inequality. If upward mobility
is very low, aversion to inequality is extreme and focused on the rich: people care
about their success relative to those at the top of society.
Keywords: evolution, inequality aversion, relative preferences
1 Introduction: social preferences and evolutionary motives
The standard economic agent cares only about his own material wellbeing, without regard to that of others. But in reality, when evaluating social outcomes, people display
social preferences which depend on the success of others. They display relative preferences, in which, holding their own wealth constant, they would prefer others to be less
wealthy. And they display aversion to inequality, in which, holding their wealth constant, they would prefer others’ wealth to be distributed more equally. They are willing
to make sacrifices to achieve these goals. These preferences have been studied experimentally and in surveys, and have important consequences for political outcomes.
In this paper I show that both inequality aversion and relative preferences can be
explained by an individual’s concern for the competition faced by his descendents.
Formally I show that these preferences follow from an individual’s concern for his fitness, i.e. his long-run number of descendents, and in this way they make sense evolutionarily. The intuition is easy to give. Historically wealth is positively correlated with
fertility, and so is an evolutionary advantage. The reason for relative preferences suggested here is that if other people become more successful, and are able to have more
children, then your descendents will face more competition and so will be fewer in
number. The reason for inequality aversion is that if current inequality increases, then
more of the population in future will be descended from those higher up in society,
and these descendents will therefore have more advantageous characteristics and so
will again pose a greater competitive threat to your descendents.
∗
cr250@cam.ac.uk
1
1.1 Relative preferences
Evolutionary success is always relative. (Penn (2003))
In a specific sense relative preferences mean that people care about their relative
wealth or success rather than their absolute wealth, and in a more general sense that
they prefer others to be less wealthy or successful.
A body of empirical literature has supported the idea of relative preferences. The
Easterlin (1974) paradox is that as aggregate incomes have grown in the developed
world, reported wellbeing has not.1 Subsequent work (Clark et al. (2008)) has explained this by finding that reported wellbeing is negatively associated with average
income of others in a country. To connect this work to relative preferences, we need
to assume some relationship between happiness and (decision) utility, which is not
implausible.
Experiments have found relative preferences more directly by asking people to rank
social outcomes. People express a preference (Solnick and Hemenway (1998)) for others to have much lower wealth, even if it means a reduction in their own wealth.2
Evolutionary competition provides a way to understand why we observe relative
preferences. More successful members of societies, who have access to greater resources, as a result have more surviving offspring. This may not happen in modern
times in developed countries, but is true of traditional societies (Kaplan and Lancaster
(2000); Clarke and Low (2001)), historically (Clark and Hamilton (2006)), and in equilibrium. If people have the evolutionary goal of maximizing the number of their descendents, this link between resources and children can relative consumption preferences: if others are less wealthy, they will support fewer surviving children, and your
children will have less competition for resources.
1.2 Aversion to inequality
There is widespread belief that inequality is a problem, and concern about increasing
wealth disparities within countries. An aversion to inequality is expressed in media
and political speech around the world, in policies and political movements, and has
been found in surveys and experiments.
This attitude is not only found in those poor who are the obvious beneficiaries of
policies which reduce inequality: it is also found in the middle classes who form the
center ground of democratic political competition. The high levels of redistribution
seen in developed democracies are hard to explain without inequality aversion, since
“the middle classes contemporaneously gain comparatively little” (Milanovic (2000))
from welfare states, since they are not the main recipients.3 Many revolutions - such
1
This body of work concerns reported wellbeing or happiness rather than preferences. But it is a
plausible conjecture that if relative success is important to happiness, relative success is also important
to utility, i.e. preferences.
2
See Clark et al. (2008) for more discussion of experimental and other evidence for relative preferences.
3
Inequality aversion as described in this paper does not explain the minority of very poor who vote
for unredistributive parties or the minority of the very rich who vote for redistributive parties - in other
words there is some unexplained heterogeneity among voters. The paper does attempt to explain why
most people in the middle in general support more redistribution than standard models with self-
2
as communist revolutions - which have targeted inequality are a more extreme case,
since they have resulted in huge economic costs to almost everyone in those societies.
It is hard to explain the attraction of these revolutions without inequality aversion.
Aversion to inequality is expressed in hypothetical social choices made in surveys
(Pirttilä and Uusitalo (2008), Amiel et al. (1999)). A negative relationship between
inequality and happiness is found in a large body of happiness research (e.g. Alesina
et al. (2004), Oshio and Kobayashi (2011)), measuring inequality at national or local
levels. On a small scale (interactions between a few individuals) inequality aversion is
also found in experimental work in humans (survey: Cooper and Kagel (2009)) and
even monkeys (Brosnan and De Waal (2003)).4 Alesina et al. (2004) suggest that perceptions of greater social mobility may reduce inequality aversion.
It is the position of the wealthiest in society that is of most concern to many people.
Media portrayals of the problems of inequality in world economies focus attention on
wealth of the very rich relative to others. The “We are the 99%” movement in 2012,
chose to target the top 1%. Many compare their lives to those of the richest, and desire
that the richest should be less rich.
In summary, large majorities regard inequality as a problem, and in particular the
relative wealth of the most wealthy. Many are willing to make sacrifices to reduce inequality, with important political consequences.
1.2.1 An explanation: inequality in fertility
One way to understand aversion to inequality, and in particular a particular concern
to reduce the wealth of those at the top, is to understand the demographic threat that
the rich and powerful can pose over generations to those below them.
The children of the most successful members are themselves relatively successful,
for reasons of nature and nurture (e.g. Bowles and Gintis (2002)). Since wealthier
members of society are able to support more children, in a stable (Malthusian) society, there must be more downward mobility than upward mobility. Clark and Hamilton (2006) confirms this theoretical prediction with data from pre-industrial England.
Those at the top have many surviving children, some of whom stay at the top, and
some of whom come down a level in society. Those at the bottom rise only rarely rise
up the social ladder, and the remainder are gradually replaced by those moving down
from higher classes. The long run population is disproportionately represented by descendents of those higher up in society, and those below must rely on the possibility of
upward mobility in a future generation.
The greatest competitive threat then to one’s descendents comes from those at the
higher up in society. If a society becomes more equal, those higher up can support
fewer children, and more of the next generation will descend of those from lower down.
These children and their descendents will pose a lesser competitive threat to your descendants. So the middle classes may be willing, if they care about their progeny, to
make sacrifices to hurt those higher up.
regarding preferences would predict.
4
This summary oversimplifies the experimental work relating to fairness, but the surveys are more
directly relevant to the setting here, since they pertain to distribution of wealth among classes, rather
than among a few individuals. Nevertheless, the psychological processes in the two settings and the
evolutionary background may be very connected.
3
The degree of inequality aversion generated by future competition depends on mobility. If mobility is “complete” and the children of those higher up are no different from
children of those lower down (probabilistic equality of outcomes) then an additional
child of any parent generates the same expected amount of competition regardless of
the parent’s attributes. If mobility is limited, inequality aversion arises and the lower
the mobility, the more time (measured in generations) those at the top have to compound their advantages. If upward mobility is very low, competition comes almost exclusively from those at the top and an extreme form of inequality aversion arises which
targets those at the top.
1.3 Contents
Section 2 sets up the basic model, where the success of children is related to that of
parents by a Markov process; this is called “natural transmission”. The main results
of relative preferences and inequality aversion are given here: relative preference, inequality aversion, and the effect of increased social mobility. Section 3 gives extreme
results (“targeting the rich”) when upward mobility is very low, in an extended model
which allows for resource transmission (which diminishes in the number of children,
giving a quality/quantity trade-off) in addition to natural transmission. Section 4 discusses the appropriateness of the equilibrium model.
2 Inequality aversion and relative preferences in a closed
society
The two models in this paper study a single closed society which does not interact with
other societies. Any competition therefore is internal rather than external.
For simplicity the approach will be to take future relative fertility as exogenous, and
consider preferences over the distribution of current fertility. Given a policy setting,
such a preference will allow us to derive preferences over policies. For example policies
which differ in the level of redistribution will affect the distribution of wealth among
different social groups, and allow them to support more or fewer children, resulting
in a distribution of fertility. Preferences over the distribution of fertility then result
in preferences over policies, which should in turn determine political activity such as
voting over such policies.
2.1 Model 1: natural transmission of qualities
A golden parent will sometimes have a silver son, or a silver parent a golden son. Plato, The Republic
There are various personal characteristics that affect the economic success of an
individual and success in supporting offspring: health, intelligence, emotional qualities, and so on. These qualities are imperfectly transmitted to children. The first model
considers characteristics that are passed from parent to child in a way that does not
4
depend on the number of children. I call this type of transmission “natural transmission”. This will include genetic transmission - of the genetic components of intelligence, health, personality, and other characteristics (nature).5 But it also include socially transmitted characteristics (nurture) provided that the transmission is independent of the number of offspring.6
Formally each parent has a combination of qualities or type described by g ∈
{1, . . . n}. Each child of a parent of type g 0 is of type g with probability H g g 0 , so that
H is the Markov transmission matrix.
There is “complete mobility” if the child’s type is unrelated to the parent’s:
Definition 1. There is complete mobility if for any g 0 , H g 0 g is independent of g .
There is “incomplete mobility” if higher types of parent are likelier to have higher
types of children, in the sense that the probability distribution of the type of a child of
a parent of type g , with mass function c → Hcg , is strictly increasing in g (FOSD):
P
Pm
Definition 2. There is incomplete mobility if for any g 2 > g 1 : m
c=1 Hcg 1 ≥ c=1 Hcg 2
for all m, with strict equality for some m.
The first assumption is that any type can be reached from any other type in a finite
chain of mutations with positive probability.
Assumption 1. For any i , j ∈ {1, . . . n}, there exists a sequence g 1 , g 2 . . . g k with g 1 = i and
g k = j where H j i > 0.
The initial population ¡is p¢0 ∈ Rn+ . Fertility in period 0 is f ∈ Rn+ and
¡ t ¢ subsequently
t
∗
∗
n
fertility in period t ≥ 1 is δ p π , where π ∈ R+ is exogenous and δ p ∈ R describes
a ¡level
¢ of competition in period t ≥ 1 based on the current population, with higher
δ p t describing less competition. So the expected number of
offspring in
¡ surviving
¢ ∗
t
generation t of a person with gene g is f g in period 0 and δ p πg in period t ≥ 1.
Higher g describes a greater ability to support children in normal settings, so π∗g is
assumed to be strictly increasing in g .
Stable population dynamics are implied by the following standard assumption that
small populations allow fast growth rates, while as population increases, survival becomes harder and an extra unit of population is not maintained in the next generation:
¡ ¢
Assumption 2. δ ¡p ¢is continuous,
decreasing,
¯ ¯differentiable on p À 0, and¡strictly
¢¡
¢
¡ ¢with
0
¯
¯
δ (0)
¡ ¢π1 > 1 and δ p πn < 1 for p sufficiently large. When δ p π · p = 1, δ p p +
δ p ≥ 0.7
These assumptions give a stable long-run population:
5
See Bowles and Gintis (2002) for estimates of how much genetic factors are responsible for the current correlation in wealth across generations.
6
Resources such as wealth, education and nutrition, are passed to children in quantities which diminish with the number of children, at least beyond a certain point. These are considered in Section
3.
7
¡ This
¢ ¡ last
¢ part of this assumption in words is that when total population remains the same
(δ p π · p = 1), an extra unit of population (with the average distribution of types) will weakly increase
the population in the next period. It is used to give that population converges to a global attractor in the
long run.
5
Lemma 1. There exists a long-run population p ∗ 6= 0 such that for any period 1 population, population subsequently converges to p ∗ .
Since we are interested in preferences over policy in generation 0, which affects f ,
we consider preferences over f . Each type g will have a different preference over f , and
this preference will display inequality aversion and relative preferences in the senses
described below.
Each person’s utility function is the expected share of her descendents in the longrun population. This is evaluated before having children, i.e. before the number of
children and their types are known. The fact there is a stable long-run population implies that this is equivalent to caring about the absolute number of long-run descendents:
Definition 3. Given f , the expected population in period t descending from a period-0
P
g is d gt . The total population is P t = g p gt .
µ t¶
³ ´
¡ ¢¡ ¢
¡ ¢¡ ¢
¡ ¢¡ ¢
dg
Let u g f := u prop g f := limt →∞ P t and u abs g f := limt →∞ d gt .
Corollary 1. u abs and u prop represent the same preferences.
P
Proof. Since limn P t = p g∗ , one utility function is a multiple of the other:
³X ´
¡ ¢¡ ¢
¡ ¢¡ ¢
u abs g f
=
p g∗ u prop g f
2.2 Conclusion 1: relative preferences
The tears of the world are a constant quantity. For each one who begins to weep
somewhere else another stops. The same is true of the laugh... It is true the population
has increased. - Beckett, Waiting for Godot
In this model preferences over the current fertility vector are relative in two senses.
They are decreasing in the fertility of others, the competitive sense of relative preferences. And they are homogenous of degree 0, the strict mathematical sense. The
latter property holds because changing the population in any stage in a way that keeps
the proportions of genes constant keeps the proportions of types in future periods the
same and the proportions descending from any type in period 0 the same.
¡ ¢¡ ¢
Proposition 1. (Relative preferences) u g f is decreasing in f h for h 6= g and is invariant to multiplying f by a positive constant.
The simplest and benchmark form of relative preferences is where an individual
cares about the proportion of her offspring in the overall population, and has utility
function:
Definition 4. Normalized fertility:
fˆ :=
f
p0 ·
6
f
If the type of each child is independent of that of her parent, this is the preference
that emerges:
¡ ¢¡ ¢
Lemma 2. (Normalized fertility preferences). Under complete mobility, u g f = fˆg .
These will sometimes be called “naive relative preferences”, i.e. the preferences that
emerge if an individual measures competition by numbers only and not also by quality.
2.3 First order and dispersion approaches to inequality
In this section we shall consider two concepts of “equality”. The notion of increased
equality that we will use in future results is a “first-order” one, namely that a more
equal distribution is created when fertility shares are taken from higher types and given
to lower types. If f is the current fertility vector, then a randomly chosen child is dep g0 f g
scended from a parent of type g with probability p 0 fˆg = P 0 . A distribution f B is
g
ph fh
then “more equal” than f A if the a randomly chosen child under f B descends from a
lower type (FOSD) of parent than a randomly chosen child under f A :
Definition 5. If f A , f B ∈ Rn+ , f B is more equal than f A if the distribution with mass
function g 7→ p g0 fˆgA FOSD8 the distribution with mass function g 7→ p g0 fˆgB .
This can be seen as a “revolutionary” form of equality as a distribution f g which is
constant in g is not the most equal distribution possible; a decreasing f g is more equal.
As in French and communist revolutions where, in the name of equality, elites not only
had their advantages removed but were actively persecuted and often killed.
A second notion is a more “liberal” one, which captures the variability in fertility in
the population:
Definition 6. If f A , f B ∈ Rn+ , f B is less dispersed than f A if fˆGA is more risky than fˆGB ,
where G is a random variable on {1, . . . n} with mass function g 7→ p g0 .
8
Recall the standard notions of First Order Stochastic dominance and increased risk (Rothschild and
Stiglitz (1970)):
Definition. Let P1 and P2 be two probability measures on R, with associated random variables X 1 and
X 2 and associated distribution functions F i (x) = Pi ((−∞, x]) = P [X i ≤ x].
1. P1 (or X 1 ) strictly First Order Stochastically Dominates (FOSD) P2 (or X 2 ) if either of the following
equivalent conditions holds:
(a) F 1 (x) ≤ F 2 (x) for all x, with strict inequality for some x.
(b) E [u (X 1 )] ≥ E [u (X 2 )] for all weakly increasing u : R → R, with strict inequality for some such
u.
2. P1 (or X 1 ) is more risky than P2 (or X 2 ) if either of the following equivalent conditions holds:
(a) E [max (x − X 1 , 0)] ≥ E [max (x − X 2 , 0)] for all x, with strict inequality for some x.
(b) E [u (X 1 )] ≥ E [u (X 2 )] for all weakly convex u : R → R, with strict inequality for some such u.
7
Both are scale-invariant notions of equality: f 0 is more equal(/less dispersed) than
f if and only if α f 0 is more equal(/less dispersed) than β f , for α, β > 0.
In “normal conditions”, when f gA and f gB are both weakly increasing, in g , the two
notions of inequality coincide. Transferring fertility shares from lower to higher types
also increases the dispersion of fertility. For example an anonymous policy, such as
a modern tax system, which does not identify the types explicitly should result in a
weakly increasing f g : higher types will not do worse than lower types since they can
mimic lower types.
Lemma 3. Suppose f A , f B ∈ Rn+ with f gA and f gB weakly increasing functions of g ∈
{1 . . . n}. Then f B is more equal than f A if and only if f B is less dispersed than f A .
The definition of increased inequality also allows for distributions f which are not
strictly increasing. In such cases the first order notion of “more equal” used here has
a political meaning of benefiting the normally poor group at the expense of the rich
group, even when this increases dispersion.
A distribution which (temporarily) gives higher types lower fertility than lower
types is by our definition more equal than one in which types have equal fertilities.
This is the “egalité” of equalizing revolutions. The models in this paper suggest that
the interest that an individual has (in terms of reduced future competition) in transferring fertility shares from those higher up in society to those lower down (without
affecting her own position) is not satiated when they are brought to the same current
level, but extended when those who are naturally higher are brought down below those
naturally lower in society. Therefore we use the first order notion of increased equality
in the results here, which allow them to extend correctly to “revolutionary” situations
where not all the available choices f g are weakly increasing in g . (Of course other notions of equality, including dispersion, may be suited to other contexts.)
2.4 Conclusion 2: aversion to inequality
Compared to the benchmark normalized fertility preference fˆg , if H is dependent on g ,
preferences display aversion to inequality. Fixing the relative fertility of any type, that
type prefers a more equal distribution:
Proposition 2. (Inequality aversion) Under incomplete mobility, if fˆgA ≥ fˆgB , and f B is
¡ ¢¡ ¢
¡ ¢¡ ¢
more equal than f A , then u g f A < u g f B .
Preferences over the fertility vector of the current generation are relative in the
sense of homogenous of degree 0 and in the sense of decreasing in others’ fertility
(Proposition 1). They differ from the normalized fertility preference, where each type
cares about the share of her fertility without regard to how it is distributed among others, by moving in the direction of being averse to inequality.
The competitive approach to inequality aversion differs from other existing ideas
in social science. For example McAdams (1992) puts egalitarianism tentatively but
without justification in the category of “positive relative preferences” where utility is
increasing in others’ success. In Proposition 2 inequality aversion is a form of “negative
relative preference”: utility is decreasing in the success of others (Proposition 1), but
decreases more in the success of the rich than in the success of the poor.
8
This¡ result
¢ ¡ ¢ cannot be simplified to “all types prefer more equality”. Total utility
0
p
u
g
f = 1, i.e. this is a constant-sum setting. So there can be no agreement
g g
among all types that one distribution is preferred to another. One way of stating the
result is that fixing the fertility share of type g , type g would prefer the shares of other
types to be distributed more equally.
In the context of redistributionary policy preferences, suppose f is determined by
P
f (r ) j
policy choice r , with f (r ) decreasing for j > i . Then higher r results in a more equal
i
f . The highest type will prefer the lowest possible r - the conclusion of the proposition
fg
does not apply since a higher r will decrease p 0 · f . One can however conclude from the
proposition that the optimal r for any type is weakly higher than the optimal r under
the benchmark preference:
Corollary 2. Suppose that R is an ordered set of redistributionary policies such that r ∈ R
results in a fertility vector f ∗ (r ), and that if r 1 , r 2 ∈ R, r 2 > r 1 , then f (r 2 ) is more equal
ˆ ) is maximized at r = r naive ,
than f (r 1 ). Under incomplete mobility, suppose that f (r
g
¢
¡ ¢¡
∗
∗
naive
.
while u g f (r ) is maximized at r = r . Then r ≥ r
³
´
naive
naive
Proof. Suppose r < r
. Then f r
is more equal than f (r ∗ ), and since
³ ˆ
´
´´
¡ ¢¡
¢
¡ ¢³ ³
f r naive ≥ f (rˆ∗ )g , Proposition 2 implies u g f (r ∗ ) < u g f r naive , contrag
¢
¡ ¢¡
dicting the assumption that u g f (r ) is maximized at r = r ∗ .
∗
f j (r )
The assumption that f (r ) decreasing for j > i implies the conditions for Corollary
i
2 are met. In this case for highest type the least equal policy (minimum r ) maximizes
both fˆn and u (n): the conclusion of Proposition 2 does not apply since a higher r will
f ∗ (r )
decrease p 0n· f (r ) . For the lowest type the most equal policy (maximum r ) maximizes
both fˆ1 and u. Intermediate types may prefer intermediate policies and Corollary 2
implies they prefer higher policies than if they used naive preferences fˆg .
2.5 Comparative statics: the effect of increased mobility
It has been suggested that greater mobility, or belief in greater mobility, leads to greater
tolerance of inequality. Alesina et al. (2004) find empirical evidence to support this.
Benabou and Ok (2001) suggests that there may be less redistribution than expected
because while a majority may have incomes below the mean, for some forms of the
random income process a majority may rationally expect to or expect their children to
have incomes above the mean in future.
The arguments here suggest that we should expect more redistribution than there
would be under pure immediate self-interest, but a more robust relationship between
mobility and tolerance of inequality is shown, based on competition. Higher mobility
reduces the advantage that an offspring of someone high in society has over someone
lower down, by equalizing the strength of future descendents and shortening the period of compounding of fitness.
The results will be shown under stronger monotone likelihood ratio assumptions:
Assumption. Higher types have MLR-higher type children:
9
H j +1,i +1
H j ,i +1
>
H j +1,i
H j ,i
H¯ has more mobility than H if the ratio
H j +1,i +1
H j ,i +1
of higher type (j+1) children to lower
type children (j) for a higher type i+1, relative to the ratio
measure of immobility) is greater than the ratio for H¯ :
H j +1,i
H j ,i
for a lower type i (a
Definition 7. H¯ has more mobility than H if
H j +1,i +1 H j ,i
H j ,i +1 H j +1,i
>
H¯ j +1,i +1 H¯ j ,i
H¯ j ,i +1 H¯ j +1,i
Definition 8. f B is MLR-more equal than f A if
f A (h)
f B (h)
is increasing in h.
We have already seen (Lemma 2) that if there is “complete mobility”, so that H g 0 g is
independent of g , then u = u naive . There is then no aversion to inequality in the sense
rel
of Proposition 2; aversion to inequality arises under incomplete mobility (Proposition
2). We can also say that the lower the mobility, the greater the aversion to inequality:
Proposition 3. Suppose H¯ and H have incomplete mobility, and that H¯ has more mo¯
bility than H . Let u represent preferences
¡ ¢ ¡under
¢ H¡ and
¢ ¡ u¯¢ preferences
¡ ¢under
¡ ¢ H ¡. Suppose
¢¡ ¢
f B is MLR-more equal than f A , and u¯ g f B ≥ u¯ g f A . Then u g f B > u g f A .
¡ ¢¡ ¢
¡ ¢¡ ¢
Suppose that u¯ g f B = u¯ g f A . Then under higher mobility H¯ , type g loses out
in the more equal policy B in terms of relative fertility compared to A, but the level of
future competition is higher under A than B : these two considerations cancel out to
give indifference. Under lower mobility H , type g loses out just as much in the more
equal policy in terms of relative fertility, but the relative future competitiveness of A
compared to B is increased further, causing g to prefer policy B under H .
3 Extreme preferences with low upward mobility
The previous model suggests that reduced mobility is associated with greater aversion
to inequality. In an extreme case with very low upward mobility, where it is very unlikely for someone to move up to the top of society, the relative fertility of those at the
top is able to compound over many generations, and so the long run population is
composed almost exclusively of descendents of those currently at the top. In this case
those at the top are the main competitive threat, and in making social judgments individuals will be concerned about their success relative to those at the top: preferences
fg
are approximately f n . This is an extreme preference not only because it targets only
one group, but also because under it all types below n, when choosing from a range
f j (r )
of redistributive options R with f (r ) decreasing for j > i , will have max R as their ideal
i
policy.
This result holds in a setting which is extended to allow for inheritance of assets
and not just inheritance of genes.
3.1 Model 2: inherited nature vs inherited assets, and the effect of
compounding
The previous model considered transmission of attributes from parents to children
which does not depend on the number of children. But some mechanisms, such as
10
inheritance of wealth, involve investments in children which diminish as the number
of children increases.
While investment in children is important, it does not form part of the explanation
of extreme inequality aversion here because differences in fertility due to inherited assets do not compound over time in the way that differences due to inherited qualities
do.9 If people were identical apart from an initial endowment of wealth, differences
in wealth would dissipate as the wealth of the wealthy would be spread over a greater
number of children.10
This difference in compounding is illustrated in this section in a model with almost
perfect natural transmission (or more precisely, where improvement in nature is unlikely). There are k generations. In a continuum population, each individual lives for
1 stage, and has type g drawn from a finite set G ⊂ R, and wealth r from a finite set R.
The initial distribution is given by p 0 ∈ 4G and an endowment profile r 0 : G → R. Individuals care either about the number or the share of their descendents after the kth
stage; again it will not matter which.
The individual chooses a level of wealth r 0 ¡for her¢ children.
The
num¢
¢ ¡ expected
¡
t
t 11
0
∗
0
t
, where
ber¡ of children
from periods 1 on is given by f g , r, r = π g , r, r κ µ , σ
¢
π∗ g , r, r 0 is strictly increasing in g , µt is the current distribution of types
and
¡
¢ret
0
0
sources,
current equilibrium strategies. In period 0, f g , r, r =
¡
¢ and
¡ σ describes
¢
λ g , r π0 g , r, r 0 , σ0 , so that λ is the effect of a current policy on the fertility of different groups.
As before there is a Markov process H on type g . In this section we study the case
when H has almost no upward mobility, in particular almost no chance of mutating
to the highest type n, and when enough highest types n remain at n to dominate the
fertility of other groups. In other words H is close to the set Λ:
¡
¢ ∗
∗
0
∗
0
Definition¡ 9. Let
Λ
be
the
set
of
H
such
that
for
every
r,
r
,
π
n,
r,
r
Hnn >
¢
maxg <n π g , r, r 0 , and for g < n, H g∗n = 0.
3.2 Conclusion: targeting the 1%
d gt
Assumption 3. The utility of type g in period t equals P k , where d gt is the expected
population in period k descending from a period-t g , and P t is the total population
P
is P t = g p gt .
Lemma 4. For any H , equilibrium payoffs are unique.
9
Some may be surprised at the statement, since some assets do increase in value at a rate of interest,
and this does compound over time. And currently wealthy people invest a greater proportion of this
wealth in assets with higher growth rates than poorer people, so that “the rich get richer and the poor
get poorer”. But 1. greater wealth may not be the cause of good investment strategies, 2. the causes of
poor investment strategies among the poor may be transmitted by “natural transmission”, and 3. over
most of human history assets (most commonly land and property) have not increased in real value.
10
What is true of the present is that the usual equilibrium correlation between wealth and fertility is
not there (see Section 4.1), and the lack of this link explains some of the increasing wealth inequality
within countries.
11
This is just a simple way of modeling a quantity/quality tradeoff. In reality many cultures children
have not received equal resources, perhaps because of indivisible goods. (Primogeniture.)
11
¡ ¢
Definition 10. The utility of g in period 0 resulting from λ is u g (λ) :=
µ
d gk
Pk
¶
.
If H is close to Λ, the long run population descends almost entirely from n types;
therefore they are the main competitive threat.
λ(g ,e (g ))
Proposition 4. If λ(n,e(n))
<
¡ ¢
u H g (λ).
λ0 (g ,e (g ))
,
λ0 (n,e(n))
¡ ¢
then for H sufficiently close to Λ, u H g (λ) >
In other words all types below n compare themselves to n, and when evaluating
current policies have (approximately) the simple objectives of increasing their own fertility and reducing that of the highest types.
This holds regardless of the initial distribution of resources, since even if those with
the highest types have the same of lower resources than those lower down, they will
still dominate in the long run since advantages of type compound and advantages of
resources do not.
Results with ¡a similar
¢ intuition would hold for more general period 0 policies, either
determining f 0 g , r, r 0 directly or affecting resources.
4 The appropriateness of equilibrium analysis
4.1 The relationship between wealth and fertility: past, present and
future
“The omission of Darwinian interpretations of contemporary reproductive patterns... reflects an omission in the whole field of human demography” R. Mace (1999).
Societies where conditions have remained stable over a long period of time show the
evolutionary (or Malthusian) equilibrium prediction that the expected number of surviving children is increasing with wealth (Kaplan and Lancaster (2000), Clarke and Low
(2001), Clark and Hamilton (2006)). This property is essential not only to the explanation of social preferences here but to even the most basic preferences such as the
desire for wealth.
The modern world (after the “demographic transition”) has two unusual properties
from the point of view of evolutionary equilibrium:
1. Technology has advanced faster than population growth, and societies have supported the poor and their children (in their own and other countries), with the
result that the poor in most countries reproduce at above replacement rates.
2. Technology and social change has given people new choices over fertility (such
as contraception vs no contraception; career vs family) and are not maximizing
fitness with respect to these unfamiliar choices (Kaplan et al. (1995)), reducing
fertility among the rich.
The consequence is that there is no longer a positive correlation between wealth and
fertility within most developed and emerging countries.
12
It is no surprise that a change in the environment will disrupt an equilibrium. This
does not mean that equilibrium analysis is not useful in studying a transition period.
Firstly current social preferences can be explained as adaptations to past (equilibrium)
conditions; just as the simple preference for greater wealth would be explained as an
adaptation to a condition where wealth is positively linked with fertility.
Secondly human fertility choices should adapt to the new environment, very
quickly in evolutionary time since the adaptations required are a simple quantitative change (more fertility) with very strong benefits to fitness, along a dimension
along which people are already heterogeneous (preferences for children, social liberalism/conservatism). By contrast the benefits of the right social preferences are weaker
under conditions where social decisions are made by large societies and so should not
change quickly.
4.2 The selection process: the evolutionary value of distributional
preferences
The model here does not study a selection process, but assumes that behavior is always
fitness-maximizing. The model here for simplicity involves a large society making decisions which affect the fertility of classes of people. An individual decision such as
voting can be expected to have very little impact, and a selection process for optimal
social choices would be very weak in this environment.
To understand how fitness-maximizing social preferences might have emerged we
can look at smaller groups, where individual social preferences do have an impact.
The logic of demographic competition that could explain inequality aversion in a small
group of hunters distributing a hunt is the same as in our setting with a continuum of
agents.
An alternative mechanism is that parents could have evolved a concern about the
“wellbeing” of children and grandchildren, where this “wellbeing” includes wealth and
social status instead of fertility, but in a way that is linked to the fertility expected of
a certain level of wealth and status. Faced with a social choice which increases inequality, parents may recognize that this will increase the competition their children
and grandchildren face, and so will reduce their status. Again the competitive logic of
this model is similar to the logic in this paper; precisely how similar depends on how
accurately parents link status and long-run fertility.
5 Conclusion
Orthodox economic agents do not display relative preferences or inequality aversion.
But both aspects of social preference have been well documented empirically. Such
preferences are not irrational in the formal sense of consistency, but are often considered irrational in a broader sense of foolish or unsophisticated, and classed with other
“behavioral” anomalies. If others are made better off, how does that affect one’s own
ability to enjoy consumption and other individual pursuits? Don’t these preferences
come from a misunderstanding that we are competing with those others, when in reality they have no power to affect our lives in the peaceful world we live in?
13
Evolutionary thinking provides a justification for such preferences. Even if we are
not competing with others now, our descendents will be. Thus in a biological sense the
social preferences described here are more rational than orthodox preferences. This
paper derives relative preferences and inequality aversion from the most fundamental
goal of an organism: reproduction and survival in the long run.
In between this goal and actual human behavior are a number of layers which implement this goal imperfectly: intermediate goods and psychological processes. Nevertheless just as rational analysis in orthodox economics has value as a benchmark, so
does fitness-maximization in providing adaptive explanations for observed behavior. I
hope that this simplified picture will help to make these well-documented social preferences more accepted by economists, and help to understand the competitive nature
of these preferences.
14
Appendix
A Proofs
 ∗

π1 0 · · ·
∗


Write Π =  0 π2
 and M = H Π. Then p t ∝ M t −1 p 1 .
..
..
.
.
A.1 Proof of Lemma 1
Theorem. (Perron-Frobenius). An irreducible non-negative matrix M ≥ 0 has a positive
eigenvalue r which is a simple root (i.e. a root of multiplicity one) of the characteristic
equation of M , and which is not exceeded by the modulus of any other eigenvalue of M .
The eigenvector corresponding to r may be taken positive µ > 0, and there are not two
linearly independent positive eigenvectors.
µ
¶
Mt p
t
t −1
It follows that p ∼ αr µ, so the proportions converge to limt P M t p
= µ.
(
)g
¯ ¯
¡ ¢
¡ ¢
By 2, for ¯p ¯ ≥ κ1 , δ p πn ≤ 1 − ²1 for some κ1 and ²1 > 0. Let κ2 := max|p |≤κ1 δ p .
¯ ¯
¯ t¯
12
¯ ¯
Then ¯p t ¯ ≤ κ2 for
© all t ≥ T ,ªfor some T . Also for some ²0∗, p ∗≥ ²0 for all t .
¡ On
¢ the line αµ : α > 0 , there is a global attractor p = α µ of the map F : p 7→
δ p M p, the unique fixed point of this map. Take a compact
interval
∞) con¡©
ª¢ I ∈ (0,
∗
k
∗
taining α . Then for any ²2 , there exists K such that F
αµ : α ∈ I ⊆ B (α , ²2 ) µ for
13
k >K.
¯ ¯
¡
¢
Since
and so uniformly
continuous
on ²0 ≤ ¯p ¯ ≤ κ2 , for d p, αµ ≤
¡ k ¡F ¢is continuous
¡
¢¢
¡
¢
¡
¢
²3 , d F p , F k αµ < ²2 and so F¡ k p ⊆¢B p ∗ , 2²2 for k > K .
¡
¢
Take T sufficiently large that d p T , αµ ≤ ²3 for some α. Then d p T +k , p ∗ < 2²2 for
k > K . So p t → p ∗ .
A.2 Algebraic expressions for utility and evolutionary values
The expected number of period t +1 type g 0 descendents of a period 1 type g is αt M gt 0 g ,
¡ ¢
Q
where αt = it =1 δ p t . The total expected number of period t + 1 descendents of a
P
current type g is αt g 0 M gt 0 g .
The expected number period 1 descendents of type g 00 of a period 0 type g is
P
f g H g 00 g and each of these has an expected number of descendents αt g 0 M gt 0 g 00 in period t + 1. So the expected number of period t + 1 descendents of a period 0 type g is
P
αt f g g 00 ,g 0 H g 00 g M gt 0 g 00 .
P
P
The population in period t + 1 is then αt g¯ p g0¯ f g¯ g 00 ,g 0 H g 00 g¯ M gt 0 g 00 .
¯
¯ ¯ ¯
¯ ¯
¯ T¯
¯ T¯
12
¯p ¯ > κ1 so that ¯p T +1 ¯ < ¯p T ¯ < M or ¯p T ¯ ≤ κ1 in which case
¯ ¯
For
¯ T +1
¯ some T ¯, pT +1 ¯≤ κ2 . Then¯ either
¯
¯p
¯ ≤ κ2 . So ¯p
¯ ≤ κ2 , and ¯p t ¯ ≤ κ2 for all subsequent t .
¡
¢
13
∗
If not, then there exists a sequence αi ∈ I and k i → ∞ with F ki αi µ ∉ ¡B (α
By
¢ , ²¡2 ) µ.
¢ taking a
n
¯ Take ²3 such that for p ∈ B (α, ²3 ) µ, F p ∈ B p ∗ , ²2 for n ≥ 0.
subsequence, we can assume αi → α.
¡
¢
¡ ¢
¯ F k¯ α∗ µ ∈ B (α, ²3 ) µ, and then for this k, F k¯ αµ ∈ B (α, ²3 ) µ
Since α∗ is a global attractor, for some k,
¡
¢
¯ contradicting that αi ∈ I and
for α close to α∗¡ by continuity
of F . Then F k αµ ∈ B (α, ²2 ) µ for k ≥ k,
¢
ki
∗
k i → ∞ with F αi µ ∉ B (α , ²2 ) µ.
15
An expression for utility is the limit of the ratio:
P
αt f g g 00 ,g 0 H g 00 g M gt 0 g 00
lim
P 0 P
t
t →∞ αt
g¯ p g¯ f g¯ g 00 ,g 0 H g 00 g¯ M g 0 g 00
P
f g g 00 ,g 0 H g 00 g M gt 0 g 00
= lim P 0 P
t
t →∞
g¯ p g¯ f g¯ g 00 ,g 0 H g 00 g¯ M g 0 g 00
¡ ¢¡ ¢
u g f
=
We can rewrite this by defining a relative evolutionary value of a period-1 type g :
P
v g := lim P
t →∞
t
g 0 Mg 0g
t
h,g 0 M g 0 h
This allows us to write utility as:
P
f g g 00 H g 00 g v g 00
¡ ¢¡ ¢
u g f =P
P
g¯ f g¯ p g¯ g 00 H g 00 g¯ v g 00
We can also define the expected relative evolutionary value of a child of a period-0
type g :
P
g 00 H g 00 g v g 00
w g := P
g¯ ,g 00 H g 00 g¯ v g 00
Then utility is:
¡ ¢¡ ¢
fg wg
u g f =P
0
g¯ f g¯ p g¯ w g¯
A.3 Motononicity of v g and w g under incomplete mobility
Lemma 5. If a, a 0 , b ∈ Rα have a i , a i0 ≥ 0, b i ≥ 0 weakly increasing, and
P
P
then a i0 b i ≥ a i0 b i .
Proof. Set a¯ ∈ Rα+1 with a¯i =
(
0
a i −1
P
aj
i =1
i >1
P
a i0 b i =
P
a¯i0 b¯ i ≥
P
0
i ≥i¯ a i
≥
P
i ≥i¯ a i ,
and similarly a¯ 0 . Then a¯i and a¯i0 are prob-
¯ So if b¯ = a¯i =
ability mass functions, and a¯ 0 weakly FOSD a.
weakly increasing, so
P
a¯i b¯ i =
P
(
0
b i −1
i =1
i >1
, then b¯ i is
a i0 b i .
Definition 11. Let D gt ,g 0 be the expected number of type g 0 descendents in period t ≥ 1
of a period 1 type g . (Here D 1g ,g 0 = 1g =g 0 .)
Claim 1. Under incomplete mobility, for any g 0 , g¯ ∈ {1, . . . n} and t ≥ 1,
strictly increasing in g .
16
P
t
g 0 ≥g¯ D g ,g 0
is
Proof. This is true for t = 1 trivially:
Suppose it holds for t = k.
π∗g 0
0
1
g 0 ≥g¯ D g ,g 0 = 1g¯ ≤g is weakly increasing in g .
P
Then π∗g 0 g 00 ≥g¯ H g 00 g 0 is weakly increasing
P
P
P
So g 00 ≥g¯ g 0 D kg ,g 0 π∗g 0 H g 00 g 0
g 00 ≥g¯ H g 00 g 0 are.
P
in
=
and
g for any g¯ since both
´
³
P
P
P
k+1
k
∗
g 0 ≥g¯ D g ,g 0 =
g 0 D g ,g 0 πg 0 g 00 ≥g¯ H g 00 g 0 is weakly increasing in g , by Lemma 5. So
¡ ¢P
P
δ p k g 00 ≥g¯ g 0 D kg ,g 0 π∗g 0 H g 00 g 0 is weakly increasing in g . So the claim holds for t =
k + 1.
Claim 2. v g is strictly increasing in g .
Proof. Claim 1, taking a limit as t → ∞, implies that the long run expected number of
descendents of a period 1 type g is weakly
in g , so v g is weakly increasing in
¡ 1 ¢ increasing
∗P 0
g . The recurrence relationship v g = δ p πg g v g 0 H g 0 g then implies that v g is strictly
P
increasing in g , since g 0 v g 0 H g 0 g is weakly increasing in g and π∗g strictly increasing.
Claim 3. w g is strictly increasing in g .
Proof. Since v g 0 is strictly increasing in g , and the distribution with mass function g 0 7→
P
H g 0 g is increasing in g (FOSD), g 0 H g 0 g v g 0 is strictly increasing in g , which implies that
w g is strictly increasing in g .
A.4 Proof of Proposition 1
A.4.1 Proof of Proposition 1
From Section A.2:
¡ ¢¡ ¢
fg wg
u g f =P
g¯ f g¯ p g¯ w g¯
¡ ¢¡ ¢
So u g f is homogenous of degree 0 in f .
A.5 Proof of Lemma 2
Since H g 0 g = H g 0 is independent of g , w g is independent of g , so:
fg wg
¡ ¢¡ ¢
u g f
=
P
=
P
17
g¯ f g¯ p g¯ w g¯
fg
g¯ f g¯ p g¯
A.6 Proof of Lemma 3
See Kakawani (1984) for a definition of Lorenz dominance and equivalence between
generalized dominance and second order stochastic dominance.
If f gA and f gB are weakly increasing, then the statement f B is more equal than f A
is equivalent to f B Lorenz dominanating f A , which is equivalent to the normalized
P
fB
p h0 f hA
Lorenz dominating
P
fA
,
p h0 f hA
which is equivalent to
P
fB
p h0 f hA
generalized-Lorenz-
fB
fA
, which is equivalent to P G0 A second order stochastically dominat0
A
ph fh
ph fh
fGB
fA
fA
P 0 A , which is equivalent to P 0 A being less risky than P 0 A since they have the
ph fh
ph fh
ph fh
dominating
P
ing
same mean.
A.7 Proof of Proposition 2
By assumption,
fˆgA ≥ fˆgB
The lottery with mass function h 7→ p h0 fˆhA strictly dominates (FSD) the lottery with
mass function h 7→ p h0 fˆhB , and w h is strictly increasing in h (Claim 3), so:
X
h
p h0 fˆhA w h >
X
h
p h0 fˆhB w h
Since utility is homogenous of degree 0:
¡ ¢¡ ¢
fˆg w g
fg wg
=P
u g f =P 0
ph fh wh
p h0 fˆh w h
Combining the two inequalities gives:
fˆgA w g
fˆgB w g
¡ ¢¡ ¢
¡ ¢¡ ¢
u g fA =P
>P
=u g f B
p h0 fˆhA w h
p h0 fˆhB w h
A.8 Proof of Proposition 3
A stronger result is proved here, in that the steady state level of inequality is allowed to
vary:
Assumption 4. π¯ ∗ has weakly lower steady-state inequality than π∗ : for all i ,
π∗i +1
π∗i
≥
π¯ ∗i +1
π¯ ∗i
The extended result is therefore that greater mobility, or lower future inequality,
makes a type more tolerant of inequality.
Definition 12. A vector x ∈ Rn is monotone if x i > 0 and x i is weakly increasing in i .
18
Definition 13. A matrix A is TP2 if A i j > 0 and
A i +1, j +1 A i , j
A i +1, j A i , j +1
> 1 for all i , j .
x i +1
xi
>
A i +1, j +1 A i , j
A i +1, j A i , j +1
>
Definition 14. If x and y are monotone vectors, y is MLR-more-equal than x if
y i +1
yi
.
Definition 15. If A and B are TP2 matrices, A is more TP2 than B if
B i +1, j +1 B i , j
.
B i +1, j B i , j +1
Lemma 6. (Karlin (1968), Lemma 1.1) A and B are TP2 matrices, then AB is TP2.
P
P
Lemma 7. If A and B are TP2 matrices with i A i j = 1 and i B i j = 1, and A is more
TP2 than B , and x is a monotone vector, then B x is MLR-more-equal than Ax.
Lemma 8. If A is a TP2 matrix, and x and y are monotone vectors with y MLR-moreequal than x, then Ay is MLR-more-equal than Ax.
¡
¢
¡
¢ ¡
¢
Proof. y x is TP2, so by Lemma 6 A y x = Ay Ax is TP2, so Ay is MLR-moreequal than Ax.
Claim 4.
v h+1
vh
>
v¯h+1
v¯h
£ ¤
Proof. Let P n be projective space.
Then [v] = lim v t , where arbitrarily v i0 = 1 and
£
¤
¯ T v¯ t .
¯ = lim v¯ t , where arbitrarily v¯i0 = 1 and v¯ t +1 = M
v t +1 = M T v t . Similarly [v]
When t = 0, v¯ t is weakly more equal than v t .
Suppose v¯ t is weakly more equal than v t . Then H¯ T v¯ t is more equal than H¯ T v t
by Lemma 8, which is more equal than H T v t by Lemma 8, so H¯ T v¯ t is more equal
¯ v¯ T t = Π
¯ H¯ v¯ T t is strictly more equal than
than H T v t by transitivity, and so v¯ t +1 = M
t +1
T t
v
= ΠH v .
It follows by taking limits that v¯ is weakly more equal than v.
Applying the map once more gives strictness.
Claim 5.
w h+1
wh
>
w¯ h+1
w¯ h
£P
¤
£
¤
£ T ¤
¯ =
H j i v j so in projective space, [w] = H T v and similarly
H¯ v¯ .
Proof. w i ∝
[w]
£
¤
T
¯
¯T
¯ = H¯ T v¯ is more
Since H is more
£ T ¤TP2 than H , H is more TP2 than
£ HT ,¤and so [w]
equal than H v¯ , which is more equal than [w] = H v .
An expression for the ratio of competition levels under H of A relative to B .
P
f hA p h0 w h
P
f hB p h0 w h
A similar expression exists for H¯ , and is lower:
Claim 6.
P
P
f hA p h0 w h
f hB p h0 w h
P
>P
19
f hA p h0 w¯ h
f hB p h0 w¯ h
f B p 0 wh
B
Proof. Normalize: Let Wh = P h B h0
fg pg wg
0
¯ h = Pf h Bp h0w¯ h . Then
and W
¯
fg pg wg
P
Wh =
P
¯ h = 1.
W
Need to show:
P f hA
f hB
P
⇔
This holds since 1.
f hA
f hB
Wh
X f hA
f hB
P f hA
Wh
Wh
>
f hB
¯h
W
¯h
W
X f hA
¯h
>
W
f hB
P
is increasing in h and 2. the mass function h → Wh is FOSD
¯ h . Point 2 is implied by MLR higher:
higher than h → W
Wh+1
Wh
>
¯ h+1
w h+1
W
¯ h , i.e. w h
W
>
w¯ h+1
w¯ h .
¡ ¢¡ ¢
¡ ¢¡ ¢
Suppose u¯ g f B ≥ u¯ g f A , i.e.
f gB w¯ g
P
f hB p h0 w¯ h
P
f hA p h0 w¯ h
⇔P
f hB p h0 w¯ h
≥
≥
f gA w¯ g
P
f hA p h0 w¯ h
f gA
f gB
Then by the previous claim,
P
f hA p h0 w h
P
f hB p h0 w h
⇔P
f gB w g
>
>
f gA
f gB
f gA w g
f hB p h0 w h
f hA p h0 w h
¡ ¢¡ ¢
¡ ¢¡ ¢
⇔u g f B > u g f A
P
A.9 Proof of Proposition 4
¡ ¢
If H ∈ Λ then u (n) f =
1
pn
¡ ¢¡ ¢
and u g f = 0 provided f n > 0.
¡ ¢
¡ ¢
Then for H close to Λ, by continuity u (n) f ≈ p1n and u g ≈ 0 for g < h. Suppose
¡ ¢
¡ ¢
g , starting with endowment e g , optimally chooses r g .
¡ ¢¡ ¢
u g f
=
≈
∝
¡ ¢¡ ¢
¡ ¢¤ u g f
£
¡ ¢
u (n) f
u (n) f
¡ ¢¢ 0 ¡
¢
¡ 0 ¡ ¢¢
· ¸ ¡
0 P 0
1 λ g , e g π g , e (n) , r (n) , σ
g Hg 0 g v g , r g
¢
¡
¢P
¡
p n0 λ (n, e (n)) π0 n, e (n) , r (n) , σ0 g 0 H g 0 n v g 0 , r (n)
¡
¡ ¢¢
λ g,e g
λ (n, e (n))
20
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